Robert & Casella. Chpt 2, Ex. 2.3

The argument being that the CLT normality is sufficiently accurate with 12 terms

(a) Show that and

For any random variable it follows from the definition of expected value and variance of the uniform distribution that and . Since are i.i.d. then we can do:

(b) Using histograms, compare this CLT-normal generator with the Box-Muller algorithm.

(c)Compare both generators with rnorm

Here, I generated samples from three different normal generators and compared them by estimated PDFs, histograms and Q-Q plots. The samples generated from CLT-normal generator lacked fitness to the tail of while the Box-Muller transformation worked quite well compared with the samples generated from the R function rnorm

The pictures are kinda crappy as you can see, so here’s the R code for them in case you’d like to reproduce them: